Document Type : Research Paper


1 MSc, Department of Industrial Engineering Faculty of Engineering, Shahed University, Tehran, Iran

2 Associate Professor, Department of Industrial Engineering, Faculty of Engineering, Shahed University, Tehran, Iran

3 Associate Professor, Department of Industrial Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran


Critical path method is one of the most widely used approaches in planning and project control. Time is considered a determinative criterion for the critical path. But it seems necessary to regard other criteria in addition to time. Besides time criterion, effective criteria such as quality, cost, risk and safety are considered in this paper. Then, the developed problem is solved as a multi-attribute decision making problem by a new extension of MULTIMOORA method. Moreover, type-2 fuzzy sets are utilized for considering uncertainties. Type-2 fuzzy sets are more flexible and capable than type-1 fuzzy sets in reflecting uncertainties. Eventually, SWARA method is developed for determining the weights of efficient criteria such as time, cost, quality, risk and safety under type-2 fuzzy environment. Finally, an applied example has been solved to illustrate the calculations and the ability of the proposed approach. Based on the example, it is clear that the longest path in terms of time criterion is not a critical path, and other influential criteria are involved in determining the critical path.
Today, in the competitive business environment, project management, planning, scheduling, and project control hold significant importance. One of the widely used and common methods in the field of project planning and control is undoubtedly the Critical Path Method (CPM). In the Critical Path Method, activity durations are predetermined. However, in the real world, many projects and activities are executed for the first time and have considerable uncertainties. Therefore, obtaining an accurate estimate of the time and resources required for activities is challenging. However, considering a single criterion, such as time, will not yield fruitful results, and other influential parameters such as risk should also be taken into account. For example, a path that carries a high level of risk may not be the critical path at present, but it may become critical in the future due to the high risk involved. For this reason, this research explores other influential criteria besides time and considers them in determining the critical path.
Materials and Methods
In this study, the problem under investigation is the determination of the critical path while considering other influential criteria in addition to the time criterion. To achieve this, multiple criteria decision-making methods are used to consider criteria such as time, cost, quality, risk, and safety in determining the critical path. Furthermore, to account for the uncertainties of the real world and incorporate expert opinions, type-2 fuzzy sets are utilized. It should be noted that the MULTIMOORA method is employed for ranking the critical paths, while the SWARA method is used to determine the weights of the influential criteria in determining the critical path. Both methods have been extended and developed in a type-2 fuzzy environment.
Discussion and Results
 Initially, the proposed method is solved considering only the time criterion. As observed, the critical path has changed, indicating the importance of other criteria in determining the critical path. Then, the proposed method is solved considering pairwise combinations of the criteria, where the time criterion is treated as a fixed criterion due to its high importance. In fact, the problem is solved considering time and cost, time and risk, time and quality, and time and risk. By increasing or decreasing each criterion, the critical path changes, demonstrating the significance of all criteria in determining the project's critical path. To determine the critical path, it is necessary to consider all criteria together. These variations in the criteria and the resulting change in the critical path clearly indicate the importance and influence of other criteria in determining the critical path.
In this article, an extension of the MULTIMOORA multi-criteria decision-making method is presented in the reference section. Additionally, Type-2 fuzzy numbers, which offer more flexibility and better representation of uncertainties compared to Type-1 fuzzy numbers, are utilized. The MULTIMOORA multi-criteria decision-making method is developed to incorporate these Type-2 fuzzy numbers. The opinions of three experts are used numerically for the time and cost criteria and linguistically as linguistic variables for the quality, risk, and safety criteria. Ultimately, the weights of the influential criteria of time, cost, risk, quality, and safety are determined using the developed SWARA method under Type-2 fuzzy environment. Finally, the most critical path is determined by considering not only the time criterion but also the influential criteria of cost, quality, risk, and safety. Based on the conducted research, a set of criteria including time, cost, quality, risk, and safety are used in this article, and additional criteria can also be added to this set.


Main Subjects

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